Problem 20 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ 2 i\left(\frac{1}{2}-i\right) $$

Step-by-Step Solution

Answer

The result is \(2 + i\).

Step 1: Distribute the Term

To start evaluating the expression, distribute the term (2i) to each part of the expression inside the parentheses: \[ 2i \left( \frac{1}{2} \right) + 2i(-i) \]

Step 2: Simplify Each Term

Now, simplify each of the distributed terms:1. For the first term: \[ 2i \left( \frac{1}{2} \right) = i \]2. For the second term: \[ 2i(-i) = -2(i^2) \]Since \(i^2 = -1\), this becomes \(-2(-1) = 2\).

Step 3: Combine Like Terms

Combine the simplified terms from the previous step:\[ i + 2 \]Rearrange it into the form \(a + bi\):\[ 2 + i \]

Step 4: Identify Final Result

The expression has been evaluated to the form \(a + bi\). Therefore, the result is:\[ 2 + i \]

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is a fundamental concept. It is denoted by the symbol \( i \), and its primary defining feature is that \( i^2 = -1 \). This seems unusual because no real number squared gives a negative result. However, \( i \) allows us to work with numbers that extend beyond the real line.
This imaginary unit is part of what allows complex numbers to encapsulate a broader range of solutions for polynomial equations.
In the context of the original exercise, \( i \) was essential in expressing and manipulating the initial complex expression. Recognizing that \( i^2 = -1 \) is vital in simplifying expressions involving powers of \( i \).

Complex Multiplication

Multiplying complex numbers involves both real and imaginary parts. This is different from multiplying real numbers, as the imaginary unit leads to interesting results.

When given an expression like \( 2i \left( \frac{1}{2} - i \right) \), you have a term with \( i \) being multiplied by both a real and an imaginary component. Distributing \( 2i \) results in two separate products: \( 2i \times \frac{1}{2} \) and \( 2i \times (-i) \).

• For \( 2i \times \frac{1}{2} \), the result is \( i \), as multiplying a real number by an imaginary unit gives another imaginary number.
• For \( 2i \times (-i) \), the calculation simplifies using \( i^2 = -1 \). This becomes \(-2(i^2) = -2(-1) = 2\), changing the product back into a real number.

Understanding how expressions with \( i \) multiply is crucial in solving complex number problems.

Distributive Property

The distributive property is a basic yet powerful tool in mathematics. It states that for all numbers \( a, b, \) and \( c \), the equation \( a(b + c) = ab + ac \) holds. This property is essential when working with expressions in parentheses.

In the exercise, the distributive property was used to tackle the expression \( 2i \left( \frac{1}{2} - i \right) \). By applying the distributive property, we multiply \( 2i \) by each term inside the parentheses separately, resulting in \( 2i \times \frac{1}{2} \) and \( 2i \times (-i) \).

• This technique allows for easier manipulation and simplification of complex numbers.
• Recognizing and applying this property helps break down complicated expressions systematically.

The distributive property is integral in handling algebraic expressions, ensuring each part is accurately accounted for in calculations.